A rough set model based on fuzzifying neighborhood systems

The notion of neighborhood systems is abstracted from the geometric notion of “near,” and it is primitive in the theory of topological spaces. Now, the notion of neighborhood systems has been extensively applied in the study of rough set. The notion of fuzzifying neighborhood systems is a fuzzification of the notion of neighborhood systems, and it is initially in the theory of fuzzifying topological spaces. Said briefly, each element x and each subset A of a universe are associated with a number in the unit interval, interpreted as the degree of A being a neighborhood of x. In this paper, a model of rough sets derived from fuzzifying neighborhood systems is developed. It is shown that this model unifies many well-known rough sets such as binary relation-based rough sets, covering-based rough sets and neighborhood system-based rough sets into one framework. The new rough sets are studied from two approaches: the constructive and axiomatic approaches. Furthermore, when the fuzzifying neighborhood system is serial, reflexive, unary, transitive, symmetric and Euclidean, then the corresponding rough sets are discussed and characterized, respectively. At last, the reduction theory of this rough set model is established.